You can have as little or as much interaction with other players as you want to.
You can complete all the stories and basic side--leveling--content by yourself. At worst, you might have to wait in line for your turn at a quest objective every now and then.
There is however quests that require to group. They are all completely optional. They are really only there to get you better gear.
If your goal is to play the stories and be done, then you're perfectly okay.
However, all level 50 ["Endgame"] content is grouped saved for dailies, which, as the name implies are simple quest chains you do ever day. They are very tiresome and tedious, and if you don't plan on doing any of the operations, not worth it.
I do encourage you to try engaging conversation though. The simplest way is to just join in some Gen chat conversations (The non trolling ones.) I know F2P have limited access to Gen chat, but I think they're at least unrestricted on the starter worlds and fleets. Someone else would know better. At the very least, you can join a guild.
Meeting people in-game can be a rewarding experience and a great way to make friends. Remember, you already know you share at least one common interest. I personally have met some of my not best "real-life" friends in MMOs. Heck, I know people who've meet their spouses.
I'm a relative MMO n00b myself (about four years). Like you, I had huge reservations when I started and never thought I'd like it. Now I wonder what the heck I was ever thinking.